Wave scattering by a permeable barrier over undulating bed topography

Abstract

The scattering of surface water waves by bottom undulation in the presence of a permeable vertical barrier is investigated for its solution. A mixed boundary value problem (BVP) arises here in a natural way while examining this physical problem. Regular perturbation analysis is employed to determine the solution of the BVP. By utilizing this analysis the given BVP reduces to two different BVPs up to first order. The solution of the zeroth order BVP is obtained with the aid of eigenfunction expansion method in conjunction with least-squares approximation. The first order BVP is solved with the help of the Green's integral theorem and the physical quantities, namely the reflection and transmission coefficients, are obtained in the form of integrals which involve the bottom undulation and the solution of the zeroth order BVP. A particular form of the bottom undulation which closely resembles to some obstacles made by nature due to sedimentation and ripple growth of sand, is considered to evaluate these integrals. The variation of these coefficients is examined for different values of the porous effect parameter, barrier length, number of ripples and ripple amplitude.